Decoherence is well-studied in the context of inflation [1]}, [2]}, [3]}.
Nelson [3]} argued that the gravitational nonlinearities (from coupling between long-wavelength fluctuations and an environmental sector in the interaction action) provide a minimal mechanism for generating classical stochastic
perturbations from inflation via the decoherence. The best-suited framework to discuss about decoherence
of cosmological perturbations and the quantum to classical transition is the Schrödinger field theory. This picture is the natural framework to study the entanglement between the fields [5]} and can be used to study the entanglement effect on curvature power spectrum [6]} and bispectrum using the interaction picture [7]}. There is no reason that one could not consider a more general initial
state such as an entangled one especially that inflation be an (low energy) effective theory of a fundamental theory such as quantum gravity or has multiple fields.
